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In the current round, I have no resources but have placed 1 meeple on a card and the rest on wood. I roll the die and get a few wood resources which can cover the cost of the card. Am I allowed to use the resources gained in the current turn to buy the card?

The start player begins. He uses all his placed people figures. Only
then does the next player, in clockwise order, use his people. The
order a player chooses to use his people is completely up to him.

And on p6, explaining how to resolve a people figure on a civilisation card:

If the player cannot or does not want to pay the required resources, he takes back his people figure and leaves the card on the game
board.

Emphasis mine. You can certainly wait to see what resources you get from other people before turning your attention to the civilisation card. And you don't even have to buy the card, even if you have by that point obtained the resources to do so.